Public events, such as concerts and sports games, can be major attractors for large crowds, leading to irregular surges in travel demand. Accurate human mobility prediction for public events is thus crucial for event planning as well as traffic or crowd management. While rich textual descriptions about public events are commonly available from online sources, it is challenging to encode such information in statistical or machine learning models. Existing methods are generally limited in incorporating textual information, handling data sparsity, or providing rationales for their predictions. To address these challenges, we introduce a framework for human mobility prediction under public events (LLM-MPE) based on Large Language Models (LLMs), leveraging their unprecedented ability to process textual data, learn from minimal examples, and generate human-readable explanations. Specifically, LLM-MPE first transforms raw, unstructured event descriptions from online sources into a standardized format, and then segments historical mobility data into regular and event-related components. A prompting strategy is designed to direct LLMs in making and rationalizing demand predictions considering historical mobility and event features. A case study is conducted for Barclays Center in New York City, based on publicly available event information and taxi trip data. Results show that LLM-MPE surpasses traditional models, particularly on event days, with textual data significantly enhancing its accuracy. Furthermore, LLM-MPE offers interpretable insights into its predictions. Despite the great potential of LLMs, we also identify key challenges including misinformation and high costs that remain barriers to their broader adoption in large-scale human mobility analysis.